If the curves $x = y^{4}$ and $xy = k$ cut at right angles, then $(4 k)^{6}$ is equal to:

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Solution

Step 1: Find the slope of tangent
If the curves $x = y^{4}$ and $x y = k$ cut at right angles.
The slope of tangent to the curve $x = y^{4}$ .
$\begin{array}{rcl} \Rightarrow 1 & = & 4 y^{3} \frac{d y}{d x} \\ \Rightarrow m_{1} & = & \frac{1}{4 y^{3}} . . . . . (1) \end{array}$
And the slope of tangent to the curve $x y = k$ .
$\Rightarrow x \frac{d y}{d x} + y = 0 \Rightarrow m_{2} = \frac{- y}{x}$
Step 2: Find value of $(4 k)^{6}$
As the curves are orthogonal so $m_{1} m_{2} = - 1$
$\Rightarrow \frac{1}{4 y^{3}} \times \frac{- y}{x} = - 1 \Rightarrow \frac{1}{4 x y^{2}} = 1 \Rightarrow \frac{1}{4 y^{6}} = 1 [∵ x = y^{4}] \Rightarrow y^{6} = \frac{1}{4}$
$\Rightarrow k = {y_{1}}^{5} \Rightarrow k^{6} = y_{1}^{30} \Rightarrow k^{6} = {(\frac{1}{4})}^{5} (4 k)^{6} = 4^{6} \times k^{6} = 4$
So, the value of $(4 k)^{6} = 4 .$

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If the curves x=y4 and xy=k cut at right angles, then (4k)6 is equal to:

If the curves $x = y^{4}$ and $xy = k$ cut at right angles, then $(4 k)^{6}$ is equal to: